1b. A few students did an epsilon-delta proof, but had some notation trouble. Others tried to argue that f was a sum of linear/constant functions, which must be continuous. Some other students attempted to show $\lim_{x -> x_0} f(x) = f(x_0)$, but did not explain every step.
2a. Most students got this, but did not conclude $\nabla g(x) = x / \|x\|$.
2b. Almost everyone thought $g(x) = \|x\|$ for this question. Only a few of students argued why $f'(t) = \nabla g(a - tb)^T (-b)$; some students just stated this as a fact, but did not explain why.
3a. No problems here.
3b. Many students thought log = log_10, but most used log = ln.
4a. Full marks were given to anyone who did this question. Glancing at the answers showed that many students do not know how to properly write sets, especially for the boundary of the first three sets. Also, almost all students would only write: "D is closed" rather than "D is not open and D is closed".
4b. Most students had difficulty with this question. Some hand-waving proofs, and even a proof by example. Lots of claims and implications were made and not explained. However, there were a some students who clearly had experience with topology proofs and would form a sequence x_n using r = 1/n.
5a. Full marks were given to anyone who did this question. Seemed well done, but I don't think many students used the fact that rank(A) = rank(A^T).
5b. Most people got the eigenvalues here, but there were some students who had trouble getting the eignvectors by hand, especially for the 3x3 matrix.
5c. Well done. Almost everyone concluded the rank is minimized when \rho = 0, however many students did not explain that if \rho \neq 0, then rank(C) = 3.